Gleichmäßig monotone Potentialoperatoren auf einigen Funktionalräumen. (Uniformly monotone potential operators on some functional spaces). (German) Zbl 0627.47011
Let B be a uniformly convex Banach space whose dual space \(B^*\) is also uniformly convex. Typical example of such space is \(L_ p(G)\) \((1<p<\infty)\) where G is a domain in Euclidean space.
Let \(\phi\) be a function from \(R_+\) into \(R_+\), where \(R_+\) is the space of non-negative real numbers, strictly increasing with \(\phi (0)=0\) and \(\phi\) (r)\(\uparrow \infty\) as \(r\uparrow \infty\). \(\phi\) is called a standard function (Eichfunktion).
The dual map B into \(B^*\) induced by \(\phi\) is given by \[ J_{\phi}(b)=\{b^*\in B^*;\quad \| b^*\| =\phi (\| b\|),\quad (b^*,b)=\| b^*\| \| b\| \}. \] Furthermore, \(J_{\phi}(b)=\Psi '(\| b\|)\) with \(\Psi (r)=\int^{r}_{0}\phi (s)ds\), \(r\geq 0\), i.e \(J_{\phi}\) is a gradient of the functional and so it is recognized as potential operator.
The author of this note considered the properties of such operators and got several lemmas and applications to \(L_ p(G)\) or Sobolev spaces. He considers also other functionals defined on a subspace of Sobolev spaces whose gradient exists and has remarkable proerties such as uniform monotoneness.
Let \(\phi\) be a function from \(R_+\) into \(R_+\), where \(R_+\) is the space of non-negative real numbers, strictly increasing with \(\phi (0)=0\) and \(\phi\) (r)\(\uparrow \infty\) as \(r\uparrow \infty\). \(\phi\) is called a standard function (Eichfunktion).
The dual map B into \(B^*\) induced by \(\phi\) is given by \[ J_{\phi}(b)=\{b^*\in B^*;\quad \| b^*\| =\phi (\| b\|),\quad (b^*,b)=\| b^*\| \| b\| \}. \] Furthermore, \(J_{\phi}(b)=\Psi '(\| b\|)\) with \(\Psi (r)=\int^{r}_{0}\phi (s)ds\), \(r\geq 0\), i.e \(J_{\phi}\) is a gradient of the functional and so it is recognized as potential operator.
The author of this note considered the properties of such operators and got several lemmas and applications to \(L_ p(G)\) or Sobolev spaces. He considers also other functionals defined on a subspace of Sobolev spaces whose gradient exists and has remarkable proerties such as uniform monotoneness.
Reviewer: S.Koshi
MSC:
47B38 | Linear operators on function spaces (general) |
46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
47H05 | Monotone operators and generalizations |
Keywords:
uniformly convex Banach space; standard function; potential operator; Sobolev spaces; uniform monotonenessReferences:
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